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Using identities Use the identity sin 2x=2 sin x cos x sin 2 to find d/dx (sin 2x). Then use the identity cos 2x = cos² x−sin² x to express the derivative of sin 2x in terms of cos 2x.
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Ch. 3 - Derivatives
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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TOC.tableOfContents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Capitolo 3, Problema 3.5.54
Verifying derivative formulas Verify the following derivative formulas using the Quotient Rule.
d/dx (csc x) = -csc x cot x
Guida verificata passo dopo passo1
Start by recalling the definition of the cosecant function: \( \csc x = \frac{1}{\sin x} \). We need to find the derivative of this function with respect to \( x \).
Apply the Quotient Rule for derivatives, which states that if you have a function \( \frac{u}{v} \), its derivative is \( \frac{v \cdot u' - u \cdot v'}{v^2} \). Here, \( u = 1 \) and \( v = \sin x \).
Calculate the derivatives: \( u' = 0 \) since the derivative of a constant is zero, and \( v' = \cos x \) since the derivative of \( \sin x \) is \( \cos x \).
Substitute these into the Quotient Rule formula: \( \frac{\sin x \cdot 0 - 1 \cdot \cos x}{(\sin x)^2} = \frac{-\cos x}{\sin^2 x} \).
Recognize that \( \frac{-\cos x}{\sin^2 x} \) can be rewritten using trigonometric identities as \( -\csc x \cdot \cot x \), thus verifying the derivative formula \( \frac{d}{dx}(\csc x) = -\csc x \cot x \).
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Quotient Rule
The Quotient Rule is a fundamental technique in calculus used to differentiate functions that are expressed as the ratio of two other functions. It states that if you have a function f(x) = g(x)/h(x), the derivative f'(x) can be calculated using the formula f'(x) = (g'(x)h(x) - g(x)h'(x)) / (h(x))^2. This rule is essential for verifying derivatives of functions like csc(x), which can be expressed as a quotient.
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06:43
The Quotient Rule
Cosecant Function
The cosecant function, denoted as csc(x), is the reciprocal of the sine function, defined as csc(x) = 1/sin(x). Understanding the properties and behavior of the cosecant function is crucial for differentiating it, as it influences the application of the Quotient Rule. Additionally, knowing the relationship between csc(x) and sin(x) helps in simplifying the derivative calculations.
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Percorso guidato
6:22Graphs of Secant and Cosecant Functions
Trigonometric Identities
Trigonometric identities are equations involving trigonometric functions that hold true for all values of the variables involved. Key identities, such as sin^2(x) + cos^2(x) = 1, are often used in calculus to simplify expressions and derivatives. Familiarity with these identities is important when verifying derivative formulas, as they can help in transforming and simplifying the resulting expressions from the differentiation process.
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7:17Verifying Trig Equations as Identities
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Domanda del libro di testo
Orthogonal trajectories Two curves are orthogonal to each other if their tangent lines are perpendicular at each point of intersection (recall that two lines are perpendicular to each other if their slopes are negative reciprocals). A family of curves forms orthogonal trajectories with another family of curves if each curve in one family is orthogonal to each curve in the other family. For example, the parabolas y = cx² form orthogonal trajectories with the family of ellipses x²+2y² = k, where c and k are constants (see figure).
Find dy/dx for each equation of the following pairs. Use the derivatives to explain why the families of curves form orthogonal trajectories. <IMAGE>
y = cx²; x²+2y² = k, where c and k are constants
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Domanda del libro di testo
27–76. Calculate the derivative of the following functions.
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Domanda del libro di testo
9–61. Evaluate and simplify y'.
y = (2x−3)x^3/2
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Domanda del libro di testo
Evaluate the derivative of the following functions.
f(x) = sin-1 2x
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Domanda del libro di testo
Calculate the derivative of the following functions. In some cases, it is useful to use the properties of logarithms to simplify the functions before computing f'(x).
f(x) = In(3x + 1)⁴
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